Question 45
Let $f : [0, 1] \to [0, 1]$ be a continuous function and $x_0 \in [0, 1]$. Define the sequence $(x_n)_{n \in \mathbb{N}}$ by
$$
x_{n+1} = \int_{0}^{\frac{1}{n+1}} (x_0 + x_1 + \dots + x_n) f(x) \, dx.
$$
Prove that the sequence $(x_n)_{n \in \mathbb{N}}$ is convergent.
$$
x_{n+1} = \int_{0}^{\frac{1}{n+1}} (x_0 + x_1 + \dots + x_n) f(x) \, dx.
$$
Prove that the sequence $(x_n)_{n \in \mathbb{N}}$ is convergent.